Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the average value of the function on the given interval. $$ f(x)=2+|x| ; \quad[-2,1] $$

Step-by-Step Solution

Answer

The average value is approximately 2.833.

Step 1: Understand the Problem

To find the average value of the function on a given interval, we use the formula for the average value of a function \[\text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx\]In this problem, the function is \( f(x) = 2 + |x| \) and the interval is \([-2, 1]\).

Step 2: Recognize the Form of the Function

The function \( f(x) = 2 + |x| \) contains an absolute value. We must consider that \(|x|\) behaves differently over negative and positive numbers. On the interval \([-2, 0]\), \(|x| = -x\), and on the interval \([0, 1]\), \(|x| = x\). We need to split the integral accordingly.

Step 3: Separate the Interval and Set Up Integrals

We'll calculate two separate integrals. 1. For \( x \) from \([-2, 0]\), the function is \( f(x) = 2 - x \).2. For \( x \) from \([0, 1]\), the function is \( f(x) = 2 + x \).Thus, the total integral is \[\int_{-2}^{1} (2 + |x|) \, dx = \int_{-2}^{0} (2 - x) \, dx + \int_{0}^{1} (2 + x) \, dx\]

Step 4: Compute Each Integral

Calculate the integrals separately:1. \( \int_{-2}^{0} (2 - x) \, dx \): \[ = \left[ 2x - \frac{x^2}{2} \right]_{-2}^{0} = \left( 2(0) - \frac{0^2}{2} \right) - \left( 2(-2) - \frac{(-2)^2}{2} \right) = 0 - (-4 - 2) = 6 \]2. \( \int_{0}^{1} (2 + x) \, dx \): \[ = \left[ 2x + \frac{x^2}{2} \right]_{0}^{1} = \left( 2(1) + \frac{1^2}{2} \right) - \left( 2(0) + \frac{0^2}{2} \right) = 2.5 \]

Step 5: Combine the Results

Add the results of both integrals to find the total integral:\[\int_{-2}^{1} (2 + |x|) \, dx = 6 + 2.5 = 8.5\]

Step 6: Divide by the Interval Length

The interval length is \( b-a = 1 - (-2) = 3 \).Use the average value formula:\[\text{Average value} = \frac{1}{3} \times 8.5 = \frac{8.5}{3}\]Therefore, the average value of the function \( f(x) \) on the interval \([-2, 1]\) is approximately \( 2.833 \).

Step 7: Final Result

The average value of the function \( f(x) = 2 + |x| \) on the interval \([-2, 1]\) is approximately \( 2.833 \).

Key Concepts

Integral Calculus

Integral calculus is a fundamental part of mathematics that deals with accumulation of quantities. It’s often used to find areas under curves. In this exercise, we are tasked with finding the average value of a function, which involves integrating the function over a specified interval and then dividing by the interval's length.

The basic formula for the average value of a function \( f(x) \) on an interval \([a, b]\) is given by:

• \[ \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]

In our problem, the function provided is \( f(x) = 2 + |x| \) and the interval is \([-2, 1]\). To compute the average value, we must first compute the definite integral over this interval. Integrating in these cases means finding the area under the curve of \( f(x) \) from \(-2\) to \(1\).

This definite integral is the total area covered by the function over the interval. Once we have this, dividing by the length of the interval (which is \(3\) in this particular case) will give us the desired average value.

Piecewise Functions

A piecewise function is a function defined by different expressions for different parts of its domain. For our function \( f(x) = 2 + |x| \), this concept plays a crucial role.

Since \(|x|\) can take different forms depending on whether \(x\) is positive or negative, we essentially need to split the function into pieces.

• On the interval \([-2, 0]\), \(|x| = -x\) because \(x\) is negative or zero.
• On the interval \([0, 1]\), \(|x| = x\) because \(x\) is non-negative.

This requires us to evaluate the integral separately on each subinterval, transforming our single integral into a piecewise integral computation.

By handling these subintervals separately:

• For \([-2, 0]\), we integrate \( f(x) = 2 - x \).
• For \([0, 1]\), we integrate \( f(x) = 2 + x \).

Each piece is carefully calculated before combining them to find the total area under the curve over the complete interval.

Absolute Value Function

The absolute value function, denoted \(|x|\), is a function that measures the distance of a number from zero on the number line, ignoring direction. It outputs non-negative numbers and is commonly used in mathematics to describe scenarios that need magnitude without sign.

For the function \( f(x) = 2 + |x| \), \(|x|\) affects the behavior of \(f(x)\) by changing its form based on the sign of \(x\):

• If \(x\) is negative, \(|x| = -x\).
• If \(x\) is zero or positive, \(|x| = x\).

Understanding this behavior is critical when integrating piecewise functions, as it influences how we divide and evaluate the intervals.

The absolute value transforms our function into two line segments, which explains why different expressions are used for different parts of the domain. Knowing how absolute values work makes tackling functions like \( 2 + |x| \) simpler, breaking them down into recognizable linear components.